Conic Section Equations

Conic sections are curves formed by the intersection of a plane and a cone. Understanding their equations—like circles, ellipses, hyperbolas, and parabolas—helps us analyze their shapes and properties, which is essential in algebra and precalculus.

1. Circle equation: (x - h)² + (y - k)² = r²

   • Represents all points equidistant from a center point (h, k).
   • r is the radius, determining the size of the circle.
   • The center (h, k) can be any point in the Cartesian plane.

2. Ellipse equation: (x - h)²/a² + (y - k)²/b² = 1

   • Describes a set of points where the sum of distances to two foci is constant.
   • a and b represent the semi-major and semi-minor axes, respectively.
   • The orientation of the ellipse depends on the values of a and b.

3. Hyperbola equation (horizontal transverse axis): (x - h)²/a² - (y - k)²/b² = 1

   • Represents two separate curves (branches) that open left and right.
   • The distance between the foci is greater than the distance between the vertices.
   • a determines the distance from the center to the vertices along the x-axis.

4. Hyperbola equation (vertical transverse axis): (y - k)²/a² - (x - h)²/b² = 1

   • Similar to the horizontal hyperbola but opens upwards and downwards.
   • The center is at (h, k), and a indicates the distance to the vertices along the y-axis.
   • The asymptotes guide the shape of the hyperbola.

5. Parabola equation (vertical axis of symmetry): (x - h)² = 4p(y - k)

   • Describes a U-shaped curve that opens upwards or downwards.
   • p is the distance from the vertex to the focus, determining the "width" of the parabola.
   • The vertex is located at (h, k).

6. Parabola equation (horizontal axis of symmetry): (y - k)² = 4p(x - h)

   • Similar to the vertical parabola but opens to the left or right.
   • The vertex remains at (h, k), and p indicates the distance to the focus along the x-axis.
   • The orientation is determined by the sign of p.

7. General form of a conic section: Ax² + Bxy + Cy² + Dx + Ey + F = 0

   • Represents all conic sections, including circles, ellipses, hyperbolas, and parabolas.
   • The coefficients A, B, and C determine the type of conic section.
   • Can be used to derive the standard forms of conic sections through completing the square.

8. Eccentricity formula: e = c/a (for ellipses and hyperbolas)

   • Measures the "roundness" of a conic section; e < 1 for ellipses, e > 1 for hyperbolas.
   • c is the distance from the center to the foci.
   • Helps classify the conic section based on its shape.

9. Directrix equation for parabola (vertical axis): x = h ± p

   • Represents the line(s) that are equidistant from the vertex to the focus.
   • Used to define the parabola's shape and position.
   • The directrix is perpendicular to the axis of symmetry.

10. Directrix equation for parabola (horizontal axis): y = k ± p

• Similar to the vertical directrix but oriented horizontally.
• Defines the distance from the vertex to the focus along the y-axis.
• Important for understanding the parabola's geometric properties.